Optimizing On-Demand Bus Services for Remote Areas

Abstract

This study proposes on-demand bus services for remote areas with low transit demand, incorporating travelers’ willingness to pay and values of time. To jointly optimize the on-demand service of overlapping bus routes, we construct a bi-level model. The upper-level model (UM) optimizes bus departure frequency in different time windows and ticket prices of on-demand services to minimize the total generalized cost, subject to travelers’ willingness to pay for on-demand services. The lower-level model (LM) calculates the probability of travelers choosing on-demand stops. A numerical analysis based on Meishan Island data in Ningbo indicates that with on-demand bus services, the total generalized cost incurred by buses and travelers can be reduced by 30.36% and 15.35% during rush and off-rush hours, respectively. Additionally, the waiting time at an on-demand bus stop is only 4.3 min during rush hours and 6.8 min during off-rush hours.

1. Introduction

Due to high land prices in central city areas, there is often a migration of populations and a redistribution of industry between urban and rural areas [1]. Suburbs, which are typically remote areas, are becoming more attractive for industrial agglomerations, with many plants, colleges, universities, and research institutes being relocated there [2]. However, these areas often have low population density and low travel demand in their early stages of development, resulting in infrequent public transit services and longer walking and waiting times for travelers compared to urban areas [3]. These challenges hinder the development of remote areas.

On-demand buses offer a viable solution to the problem described above. Travelers who require more flexibility and are time-sensitive are often willing to pay higher ticket prices for on-demand services, as it can result in shorter waiting times and reduced walking distances [4]. Examples of such services include taxis and online car-hailing [5], which are typically available in city centers but not in remote areas. Consequently, travelers in remote areas often face long waiting times for these services. The traditional “bus + taxi” service model in remote areas is not able to meet the diverse needs of travelers, resulting in a significant loss of market share for bus operators.

Under the above context, this paper proposes an on-demand bus model that takes into account the spatial and temporal distribution of travel demand in remote areas as well as the willingness of travelers to pay for on-demand services. This mode aims to address the difficulties faced by travelers traveling in remote areas. Furthermore, recognizing the variability in travel demand across different time windows and partially overlapping bus routes, this paper proposes a joint optimization of the frequency of bus departures and ticket prices for on-demand services. The objective of this optimization is to minimize the operational cost of buses and the time cost incurred by travelers in remote areas.

The remaining sections of this paper are organized as follows. Section 2 conducts a literature review. Section 3 presents a problem description. Section 4 constructs the bi-level model. Section 5 presents a case analysis, and Section 6 summarizes the main conclusions and provides policy recommendations.

2. Literature Review

As early as the 1960s, Cole and Merritt explored possible modes of transportation services to provide convenient travel services for residents of low-density travel areas. One of the systems they proposed was an on-demand transit (ODT) system that combines the characteristics of conventional bus and taxi systems, which can be considered as the epitome of on-demand bus service [6]. Subsequently, the United States enacted the Americans with Disabilities Act in the 1990s, which stipulates that public transportation-related agencies must provide travel services for specific groups such as people with disabilities, further promoting the development of ODT [7]. Since then, scholars have studied ODT services, and the existing literature on ODT mainly covers three aspects: the characteristics and advantages of on-demand transit, bus frequency, and ticket price.

In the study of the characteristics and advantages of ODT, Daganzo proposes the on-demand travel mode and demonstrates its effectiveness in low-density areas in the study of ODT features and advantages [8]. Schasché et al. conduct a literature review on ODT and conclude that on-demand service could effectively address the difficulties and inefficiencies in public transit service in rural areas [9]. Li et al. introduce a flexible ODT and develope a public transit scheduling model to meet public transit travel needs. The results show that the flexible ODT service can reduce operating costs by 9.5% and running time by 9% [10]. Nourbakhsh and Ouyang compare the performance of flexible route transit and traditional transit for different demand levels and found that the former usually has the lowest system cost when the demands mild [11]. Mageean and Nelson evaluate ODT in urban and rural areas in Europe based on order service and route planning flexibility, and investigate the reasons for its increasing popularity [12]. Davison et al. survey ODT providers and find that they mostly use small vehicles to save operating costs [13]. Schlueter et al. analyze the challenges of on-demand public transit in rural areas using data from 38,000 trips in rural areas in Germany, and find that ODT services could improve mobility in such areas [14]. Tellez et al., Molenbruch et al., and Diana et al. point out that due to the high operating cost of the ODT system, it is mostly used as a supplement to conventional public transit in areas with low public transit coverage [15,16,17]. These research works indicate that ODT is characterized by small-sized vehicles, un-fixed routes, and short departure times, making it primarily suitable for areas with lower travel demand.

In studies on ODT departure times, Chen et al. and Wu et al. propose a comprehensive optimization method that considers bus routes, departure times, and stopping sites to enhance traveler accessibility and minimize bus operating costs [18,19]. Xiong et al. develop flexible ODT routes and optimize bus routes based on departure times for improved service [20]. Wu et al. address a routing problem of ODT with time-dependent travel time and late customers and proposed a periodic optimization approach to collect traveler demands and optimize bus routes for a given period. The numerical results indicate that a wider time window allowed for serving more travelers and lowering bus operating costs [21]. Tong et al. formulate an optimization model based on multi-commodity network flow, which not only optimizes bus capacity but also travel routes and schedules of public transit to generate long-term profit for bus companies while meeting specific traveler constraints [22]. Wang et al. propose a two-step coordinated optimization method that considers both scheduled and real-time travel demands for optimizing bus routing and departure times [23]. Sun et al. formulate a mixed-integer nonlinear model for optimizing bus timetables, fleet size, and bus routing to boost bus utilization and reduce bus operating costs [24]. Azadeh et al. propose a mixed-integer linear problem to integrate ODT and conventional bus vehicles’ departure times and travel routes, demonstrating that the integrated public transit network enhances the public transit system’s service [25]. Gkiotsalitis and Stathopoulos optimize the bus schedule to improve ODT service quality and attract more travelers [26]. Wang et al. and Kim et al. jointly optimize the departure interval and the area covered by bus stops to reduce ODT operating costs [27,28]. Li et al. propose an opportunity charging strategy for electric ODT and coordinated charging plans with flexible bus service scheduling, reducing bus operating costs by 11% compared to full charging strategy [29]. Shen et al. propose an application-based ODT system design framework for optimizing bus stops, departure frequencies, and other elements and have applied it in Qingdao, China. The results show that the modified ODT system provides better public transit services [30]. This research indicates that flexible scheduling can enhance travel demand and achieve lower operating costs for ODT services. However, the difference in travel demand distribution in different time windows is not considered, and the departure frequency optimization of some overlapping routes is not explored.

In the realm of ODT ticket pricing, Liu et al. investigate the impact of ticket prices on public transit ridership and find that reducing ticket prices for normal bus service could increase public transit ridership and lead to a gain in overall revenue. However, the effectiveness of ticket price reductions varies among user groups, and factors such as the population density, destination accessibility, and distance to CBD also affect public transit ridership [31]. Kaddoura et al. analyze the relationship between ticket prices and the travel distance and determine bus ticket prices through a microsimulation of user interactions [32]. Li et al. and Amirgholy and Gonzales study the effect of normal bus ticket pricing on ODT ticket prices and conclude that ODT should consider social welfare to attract travelers to achieve overall benefits when normal bus operators set high prices. Conversely, ODT can adopt higher prices to maintain overall benefits when normal bus operators aim to maximize social welfare [33,34]. Liu et al. develop an elastic demand framework to optimize the ODT fleet size and ticket price to maximize public transit company profits [35]. Kamel et al. propose a time-based ticket price optimization method and evaluate people’s responses to changes in ticket prices, finding that optimal time-based ticket prices could help spread transit demand to off-rush hours [36]. Li et al., propose a differentiated ticket pricing strategy based on bus routes and time to improve the profit and traveler utility of the public transit system [37]. Kim and Schonfeld, Chen et al., and Guo et al. focus on maximizing social welfare through ticket price optimization [38,39,40]. Khattak and Yim and Nyga et al. study travelers’ propensity to use ODT services and their willingness to pay, finding that travelers are willing to pay more for more flexible travel services [41,42]. Ticket pricing is a crucial determinant influencing travelers’ modal choice. However, existing studies have not considered the need for jointly optimizing ticket prices and departure frequency across distinct time intervals. Moreover, such studies overlook the impact of travelers’ willingness to pay on intermodal transfers, which directly affects the service quality provided by operators.

In conclusion, the existing literature on the departure time and ticket price for ODT often fails to account for the effects of variations in traveler demand across different time periods, as well as the potential impact of several bus routes that may overlap in certain areas on the departure frequency. On this basis, this study introduces the concept of the time window to the ODT optimization problem, which incorporates factors such as travelers’ willingness to pay for on-demand services, as well as the unique characteristics of overlapping bus routes. By jointly optimizing the departure frequency across different time windows and the ticket prices for on-demand services, this study presents an optimal operation plan for public transit in remote areas.

3. Problem Description

On-demand bus services in remote areas (ODB-RA) operate through both regular and on-demand stops along designated bus routes. As shown in Figure 1, the area within the arc signifies remote regions, while the area outside the arc represents urban zones. In the event of travelers requiring on-demand services, buses will head to the on-demand stops to pick them up. The walking distance for travelers to the on-demand stop is shorter compared to normal bus stops. The ODB-RA decision-making process involves determining the bus departure frequency in different time windows and ticket prices for on-demand services. The objective is to minimize the total generalized costs incurred by both the bus operators and the travelers. The cost of bus operators includes the depreciation of bus fixed assets, labor cost, and fuel cost. Meanwhile, the cost of travelers comprises the value of their access/egress walking time and waiting time at the bus stop. The departure frequency affects the number of needed bus vehicles and then affects the labor and fuel costs of bus companies.

Moreover, the overlapped sections of bus routes, namely a part of a route that overlays with a part of some other routes, are often seen in transit network. In the case where travelers boarding at on-demand stops are headed towards destinations on the overlapped routes, all buses on the overlapping routes can provide service to these travelers, as shown in Figure 2. Since the arrival times of travelers are random, it is necessary to determine the number of travelers during different time windows. Based on this, the probability of a bus detouring to an on-demand stop is obtained, and the number of bus trips to the on-demand stop and the probability of travelers choosing different routes at an on-demand stop can be calculated based on routes’ departure frequencies.

4. Method

4.1. Model Assumptions

Assumption 1. 

According to travelers’ preference to choose the nearest stop to take the bus to their destination [43], we assume that all travelers choose the nearest bus stop to board the bus.

Assumption 2. 

Due to differences in ticket prices between normal services and on-demand services, travelers who pay the normal ticket price can only board the bus at the normal stop.

Assumption 3. 

As we are studying remote areas, it is noteworthy that many individuals have regular routines in their daily lives, such as commuting to work or school. Therefore, we assume that the OD traffic flow remains unchanged with/without bus optimization according to Chu et al. [44].

Assumption 4. 

Since we only consider the boarding demands of the on-demand stops when studying on-demand buses, we assume that all travelers get off at normal bus stops.

Assumption 5. 

Due to the fact that public transportation mainly adopts a fixed ticket price policy, we assumed that the ticket price for different bus services is fixed and does not depend on the travel distance or travel time of travelers.

4.2. Mathematical Model

The bi-level model’s structure is shown in Figure 3. The upper model focuses on optimizing on-demand service in remote areas by determining the departure frequency for various time windows and the ticket price for on-demand service, considering travelers’ willingness to pay. Meanwhile, the lower model is a Logit model to describe the mode choice behaviors of travelers, and calculates the number of travelers choosing normal stops or on-demand stops. An iterative computation between the upper and lower models results in the optimal on-demand service departure frequency and ticket price.

4.2.1. Set Definition

$i(i\in I)$ represents a normal bus stop; $d(d\in D)$ represents the traveler’s destination; $s(s\in S)$ represents an on-demand bus stop; $l(l\in L)$ denotes the bus route number; $o(o\in O)$ refers to the origin of traveler’s trip; and $w(w\in W)$,$W=\{7,8,9,10,11,12,13,14,15,16,17,18,19\}$ represents the time window. For instance, w = 7 indicates the time window between 7:00 and 8:00.

4.2.2. Upper Model: Optimization of On-Demand Service

The objective of minimizing the total generalized cost is achieved through Equation (1):

$$MIN:C_g=C_{bus}+C_{traveler}$$

(1)

where $C_g$ is the total generalized cost (Unit: RMB); $C_{bus}$ is the operation cost of bus service (Unit: RMB); and $C_{traveler}$ is the time cost of travelers (Unit: RMB), which can be calculated by Equations (2)–(4):

$$C_{traveler}=\omega (T^{walk}+T^{wait})$$

(2)

$$T^{walk}={\displaystyle {\sum }_o{\displaystyle {\sum }_i{\displaystyle {\sum }_s{\displaystyle {\sum }_d{\displaystyle {\sum }_w{\lambda }_{odwi}\times t_{oi}^{walk}+}}}}}{\lambda }_{odws}\times t_{os}^{walk}$$

(3)

$$T^{wait}={\displaystyle {\sum }_o{\displaystyle {\sum }_i{\displaystyle {\sum }_s{\displaystyle {\sum }_d{\displaystyle {\sum }_w{\lambda }_{odwi}\times t_{idw}^{wait}+{\lambda }_{odws}\times t_{sdw}^{wait}}}}}}$$

(4)

where $\omega$ is the unit time value of travelers before boarding the bus (Unit: RMB/h), including the walking time to the stop and the waiting time at the stop; $T^{walk}$ and $T^{wait}$ represent the walking time and waiting time for all travelers who board at both normal stops and on-demand stops (Unit: h); $t_{oi}^{walk}$ and $t_{os}^{walk}$ denote the walking time (Unit: h) from the origin o to the nearest normal stop i and to the nearest on-demand stop s, respectively, as shown in Equations (5) and (6); $t_{idw}^{wait}$ and $t_{sdw}^{wait}$ represent the average waiting time (Unit: h) of travelers at the normal stop i and the on-demand stop s, respectively, during the time window w, as shown in Equations (7) and (8); and ${\lambda }_{odwi}$ and ${\lambda }_{odws}$ are, respectively, the average number of travelers waiting at normal stop I and on-demand stop $s$ who travel from origin o to destination d in time window $w$, as shown in Equations (9) and (10).

$$t_{oi}^{walk}=\frac{r_{oi}^{walk}}{v^{walk}}$$

(5)

$$t_{os}^{walk}=\frac{r_{os}^{walk}}{v^{walk}}$$

(6)

$$t_{idw}^{wait}=\frac{1}{2{\displaystyle {\sum }_l{\ell }_{li}\times {\ell }_{ld}\times y_{lw}}}$$

(7)

$$t_{sdw}^{wait}=\frac{1}{2{\displaystyle {\sum }_l{\ell }_{ls}\times {\ell }_{ld}\times y_{lw}}}$$

(8)

$${\lambda }_{odwi}=Q_{odw}\times p_{1\_odw}^{}\times {\epsilon }_{oi}$$

(9)

$${\lambda }_{odws}=Q_{odw}\times p_{2\_odw}\times {\epsilon }_{os}$$

(10)

$${\ell }_{li}=\{\begin{array}{l}1, \mathrm{route} l \mathrm{passes} \mathrm{through} \mathrm{normal} \mathrm{stop} i;\\ 0, \mathrm{otherwise};\end{array}$$

(11)

$${\ell }_{ls}=\{\begin{array}{l}1, \mathrm{route} l \mathrm{passes} \mathrm{through} \mathrm{on}-\mathrm{demand} \mathrm{stop} s;\\ 0, \mathrm{otherwise};\end{array}$$

(12)

$${\ell }_{ld}=\{\begin{array}{l}1, \mathrm{route} l \mathrm{passes} \mathrm{through} \mathrm{stop} d;\\ 0, \mathrm{otherwise};\end{array}$$

(13)

$${\epsilon }_{oi}=\{\begin{array}{l}1, \mathrm{the} \mathrm{nearest} \mathrm{normal} \mathrm{stop} \mathrm{to} o \mathrm{is} \mathrm{normal} \mathrm{stop} i;\\ 0, \mathrm{otherwise};\end{array}$$

(14)

$${\epsilon }_{os}=\{\begin{array}{l}1, \mathrm{the} \mathrm{nearest} \mathrm{on}-\mathrm{demand} \mathrm{stop} \mathrm{to} o \mathrm{is} \mathrm{on}-\mathrm{demand} \mathrm{stop} s;\\ 0, \mathrm{otherwise};\end{array}$$

(15)

where $r_{oi}^{walk}$ and $r_{os}^{walk}$, respectively, represent the walking distance (Unit: km) from origin o to normal stop i and on-demand stop $s$; $v^{walk}$ represents the average walking speed (Unit: km/h) of travelers; $y_{lw}$ represents departure frequency (Unit: times/hour) of route l during the time window $w$; $Q_{odw}$ represents the number of travelers between OD pairs during different time windows; $p_{1\_odw}$ and $p_{2\_odw}$ represent the probabilities of travelers traveling between different OD pairs during different time windows choosing normal stops and on-demand stops, respectively; and Equations (11)–(15) are binary variables.

The constraints on the ticket price of on-demand services and departure frequency are shown in Equations (16)–(18).

$$0<f_1<f_2<f_1+\max (WT{P}_{f_1\to f_2})$$

(16)

$$WT{P}_{f_1\to f_2}=(U_{2\_odw}-U_{1\_odw})\times (\partial U_{1\_odw}/\partial f_1)$$

(17)

$$0\le y_{lw}\le y_{lw}^{\max }$$

(18)

where $f_1$ and $f_2$ are the ticket prices (Unit: RMB) of normal bus services and on-demand bus services, respectively; $WT{P}_{f_1\to f_2}$ is the willingness of travelers to pay for shifting from normal stop to on-demand stop; $U_{1\_odw}$ and $U_{2\_odw}$ represent the utilities of travelers choosing between normal stops and on-demand stops when traveling from origin o to destination d during time window w; and Equation (16) means that the ticket price of on-demand service is higher than that for using normal stop, and less than the maximum price a traveler is willing to pay for on-demand service. Equation (18) presents a constraint on the departure frequency, where $y_{lw}^{\max }$ denotes the upper limit of the departure frequency. In remote areas where population density is low, the frequency of bus departures is considerably lower than in densely populated urban areas. To account for this, we set the departure frequency of buses in urban areas as the upper limit of departure frequency in remote areas. Typically, the departure frequency of urban buses is 12–15 times/h, so we set $y_{lw}^{\max }$ to 15 times/h.

${\lambda }_{odwsl}$ represents the average number of travelers who choose the bus route l at the on-demand stop $s$ for travel between origin o and destination d during time window w, as shown in Equation (19):

$${\lambda }_{odwsl}={\lambda }_{odws}\times \frac{{\ell }_{ls}\times {\ell }_{ld}\times y_{lw}}{{\displaystyle {\sum }_l{\ell }_{ls}\times {\ell }_{ld}\times y_{lw}}}$$

(19)

$p_{lsw}$ represents the probability that travelers at on-demand stop s will send an order to route l during different time windows, and its calculation method is shown in Equation (20):

$$p_{lsw}=1-e^{-\frac{{\displaystyle {\sum }_o{\displaystyle {\sum }_d{\epsilon }_{os}{\lambda }_{odwsl}}}}{60y_{lw}}}$$

(20)

The operation cost of bus is shown in Equation (21):

$$C_{bus}={\displaystyle {\sum }_l{\displaystyle {\sum }_w\left\{⎣{\displaystyle {\sum }_s({\ell }_{ls}\times p_{lsw}\times d_{ls})+d_l}⎦\times y_{lw}\right\}}\times c_1+{\eta }_l\times (c_2+c_3)}$$

(21)

where $d_{ls}$ refers to the detour distance of the bus route l when passing the on-demand stop s (Unit: km); $d_l$ represents the length of the bus route l without including any on-demand stops (Unit: km); $c_1$ represents the cost of fuel per unit distance of the bus; ${\eta }_l$ represents the number of bus vehicles required for route l; $c_2$ represents the value deterioration of a bus vehicle; and $c_3$ represents the labor cost for operating a bus.

The number of bus vehicles needed for route l is calculated in Equation (22):

$${\eta }_l=\lceil \max (y_{lw})\times (\frac{d_l}{v_l}+\frac{t^{rest}}{60})\rceil$$

(22)

where $v_l$ denotes the average travel speed of bus vehicles on route l (Unit: km/h); $t^{rest}$ represents the stop time of a bus at the starting stop after the end of a journey (Unit: min); $\max (y_{lw})$ represents the maximum departure frequency for different time windows.

4.2.3. Lower Model: Bus Stop Choices

The lower model determines the stops where travelers board the bus, $p_{1\_odw}$ and $p_{2\_odw}$ represent, respectively, the probabilities of travelers choosing normal stops and on-demand stops for different time windows and OD pairs, as shown in Equations (23) and (24):

$$p_{1\_odw}=\frac{e^{U_{1\_odw}}}{e^{U_{1\_odw}}+e^{U_{2\_odw}}}$$

(23)

$$p_{2\_odw}=1-p_{1\_odw}$$

(24)

where $U_{1\_odw}$ and $U_{2\_odw}$ represent the utilities associated with selecting normal stops and on-demand stops between the origin o and destination d in time window w, as shown in Equations (25) and (26):

$$U_{1\_odw}={\alpha }_1{f}_1+{\alpha }_2{\displaystyle {\sum }_i({\epsilon }_{oi}\times t_{oi}^{walk}})+{\alpha }_3{\displaystyle {\sum }_i({\epsilon }_{oi}\times t_{idw}^{wait})}+{\alpha }_4$$

(25)

$$U_{2\_odw}={\alpha }_1{f}_2+{\alpha }_2{\displaystyle {\sum }_s({\epsilon }_{os}\times t_{os}^{walk})+{\alpha }_3{\displaystyle {\sum }_s({\epsilon }_{os}\times t_{sdw}^{wait})}}+{\alpha }_4$$

(26)

where ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$ and ${\alpha }_4$ are parameters to be estimated.

4.3. Algorithm Design

Due to the high level of interdependence between variables in the bi-level model presented in this paper, standard exact solution algorithms and solvers are not suitable for directly resolving the problem. To address this, a genetic algorithm is employed to solve the model. Chromosome encoding comprises two parts. The first part consists of 13 genes that represent the departure frequency of an on-demand bus in various time windows. (13 time windows are set in the case study section according to the actual situation). The second part consists of one gene that represents the ticket price of on-demand services, and both parts are encoded as real numbers. The fitness function is defined as the reciprocal of the upper-level model objective function (Equation (1)), meaning that the fitness function value represents the reciprocal of the generalized cost. The specific solution process is described below:

Step 1:

Set parameters such as the maximum number of evolutionary generations G, crossover probability $p_c$, and mutation probability $p_{{}_m}$.

Step 2:

Randomly generate an initial population of size N that satisfies the constraint conditions (Equations (16)–(18)).

Step 3:

Use the lower model (Logit model) to calculate the probabilities of travelers choosing different stops for travel for each chromosome under the corresponding strategy.

Step 4:

Use the results obtained from Step 3 as an input to the upper model and solve it. Calculate the fitness of all chromosomes in the population and record the individual with the highest fitness score.

Step 5:

Use the roulette wheel selection strategy to select paternal chromosomes and generate new individuals through a crossover operation with a predefined probability $p_c$. Subsequently, update the population by replacing the paternal chromosomes with the newly generated individuals. The crossover operation is illustrated in Figure 4, using the 8th gene of parental chromosomes as the crossover point to exchange the genes within the red box of the two parental chromosomes.

Step 6:

Each individual resulting from the crossover operation has a probability of $p_{{}_m}$ to undergo a mutation. Subsequently, randomly selected genes undergo mutations, generating new chromosomes. Finally, the old corresponding chromosomes are replaced with the new ones to update the population. The mutation operation is illustrated in Figure 5; select the 9th and 14th genes for mutation. The value of the 9th gene changes from 3 to 2, and the value of the 14th gene changes from 2.4 to 3.2.

Step 7:

Select chromosomes from the new population that satisfy the constraint conditions. Determine if the population size equals N. If not, return to Step 2 to continue generating new individuals. If the population size equals N, proceed with Step 3 and repeat the process until the maximum number of evolutionary generations G is reached.

Step 8:

Output the final result.

5. Case Study

5.1. Data

This paper presents a case study of the Meishan District in Ningbo City, located 25 km from the urban area. The district is characterized by a low population density, a limited public transport infrastructure, and a dependence on public transportation for shopping and commuting. The infrequency of bus departures often leads to wait times exceeding 20 min for residents. The study is focused on two bus routes which partially overlap in the region. We utilize the MapInfo software to project and overlay the geographic information of the Meishan bus stops, which are obtained through the Baidu API, onto the pre-registered Meishan regional map. Based on this, we delineate the traffic areas and road networks to generate a detailed map, which is presented in Figure 6.

In October 2022, a questionnaire survey was conducted in the Meishan area of Ningbo City to elucidate travelers’ choices of stops under different combinations of walking times, waiting times, and ticket prices. The survey aimed to collect data on residents’ gender, educational background, age, and stop choices. The detailed survey questionnaire is presented in Appendix A. We employ the formula for calculating the minimum sample size to determine the necessary sample size for this study [45], as shown below:

$$n_1=\frac{z^2{p}_1(1-p_1)}{{\Delta }^2}$$

(27)

where $n_1$ represents the minimum sample size; $\Delta$ represents the range of sampling error, which is set to 0.1 according to the research by Singh and Masuku [46]; z represents the value corresponding to the confidence interval, which is set at 1.96 (95% confidence interval) in this study; and $p_1$ is the percentage occurrence of a state or condition, which is set to 0.5 according to the research by Taherdoost [45].

We distributed a total of 300 questionnaires and collected 273 valid questionnaires (with an effective recovery rate of 91%), which is much larger than the minimum sample size calculated by Equation (27) of 97. This indicates that the sample size of the questionnaire is appropriate for our research. The questionnaire results indicate a relatively high proportion of males in the region, with 73% aged between 18 and 59 years. The specific descriptive statistics of the data are presented in

Table 1. Descriptive statistics.

Variable	Ratio
Gender	Male	52%
	Female	48%
Educational background	Junior college or below	44%
	Undergraduate	34%
	Postgraduate	22%
Age	(0,18]	12%
	(18,59]	73%
	60 and above	15%

. The lower model parameters were calibrated using the data collected from the questionnaire survey through the NLOGIT 4.0 software, which has the estimation and analysis function of a polynomial selection model [47]. The calibration results are shown in

Table 2. Calibration results of parameters.

Parameters	Coefficient	Significance
${\alpha }_1$	−0.312	0.0139
${\alpha }_2$	−0.072	0.0156
${\alpha }_3$	−0.078	0.0350
${\alpha }_4$	2.739	0.0228

. Each parameter’s significance is less than 0.05, indicating that they all have a significant impact on the traveler’s stop choice. The negative coefficients of ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$ imply a negative correlation between the utility of bus stop choices and ticket prices, walking time, and waiting time.

To understand the arrival pattern of travelers at bus stops and estimate the arrival distribution in different time windows, we conduct a survey of the number of travelers boarding at normal stops on two bus routes at various times. The survey period from 7:00 to 20:00 is divided into 13 continuous time windows, and we record the number of travelers boarding buses at bus stops in each time window. The results are shown in Figure 7.

Other parameters are shown in

Table 3. Parameter setting.

Parameter	Value	Unit	Parameter	Value	Unit
$f_1$	1	RMB/person	$v^{walk}$	4.68	km/h
$c_1$	1.44	RMB/km	$v_l$	48	km/h
$c_2$	10	RMB/h/vehicle	$t^{rest}$	10	min
$c_3$	10.8	RMB/h/person	$\omega$	10.8	RMB/h

Note: $\omega$ = the average wage of Ningbo City/22/8; $c_3$ = the average wage of Ningbo City/22/8. (Assuming a work month consisting of 22 days and a standard workday of 8 h). Data source: Ningbo Statistical Yearbook 2020 [48].

.

5.2. Bus Operation Scheme

The fitness value in the genetic algorithm is obtained by taking the reciprocal of the objective function of the upper model. The genetic algorithm is implemented using MATLAB2020a software on a Win10 64-bit computer with 8 GB of memory, and the parameter settings are N = 30, G = 300, p_c = 0.9, and p_m = 0.05.

Table 4. Optimal operation scheme.

Route	TP-DS	Departure Interval (Min/Shift)
Route a	3.6	7:00–8:00	8:00–9:00	9:00–10:00	10:00–11:00	11:00–12:00	12:00–13:00	13:00–14:00
		15	16	25	25	25	25	25
		14:00–15:00	15:00–16:00	16:00–17:00	17:00–18:00	18:00–19:00	19:00–20:00
		25	25	25	20	20	27
Route b	3.6	7:00–8:00	8:00–9:00	9:00–10:00	10:00–11:00	11:00–12:00	12:00–13:00	13:00–14:00
		20	24	30	30	30	25	30
		14:00–15:00	15:00–16:00	16:00–17:00	17:00–18:00	18:00–19:00	19:00–20:00
		30	30	30	21	26	33

Note: TP-DS is the ticket price of on-demand service.

shows the solutions of the calculation, and Figure 8 shows the bus routes. When travelers require on-demand service, the corresponding bus will detour to the on-demand stop before continuing along the normal route. However, if there is no demand at the on-demand stop, the bus will follow the normal route without detours.

By using the solution of the ticket price for the on-demand service and departure interval, we can obtain the percentage of travelers who choose the on-demand service in the traffic area of the on-demand stop during different time windows. During rush hour (7:00–8:00) and off-rush hour (11:00–12:00), the percentages are 73.75% and 63.64%, respectively. It is evident that compared to normal stops, travelers are more inclined to choose the on-demand stop, particularly during rush hour when both routes have short departure intervals under the on-demand bus mode. Furthermore, during off-rush hours (11:00–12:00), the percentage of travelers choosing the on-demand stop also reaches 63.64%. To further analyze the performance of the on-demand service, we calculate and compare the results of the operation of the transit mode without on-demand service (i.e., only with normal bus) and the on-demand bus mode (i.e., with normal bus and on-demand bus) during different time windows, as shown in

Table 5. Bus operation results.

Cost	Without On-Demand Service Transit Mode				On-Demand Bus Mode
	7:00–8:00		11:00–12:00		7:00–8:00		11:00–12:00
	Route a	Route b	Route a	Route b	Route a	Route b	Route a	Route b
TCT	1350	1139.4	675	648	867.26	699.08	562.23	491.76
OCB	109.92	73.28	109.92	73.28	168.6	126.32	131.92	89.12
TGC	1459.92	1212.68	784.92	721.28	1035.86	825.4	694.15	580.88
NBV	3	2	3	2	4	3	3	2
TPI	300	211	150	120	395.2	279.6	203	158

Note: TCT is the time cost of all travelers (RMB). OCB is the operation cost of bus (RMB). TGC is the total generalized cost (RMB). NBV is the number of bus vehicles. TPI is the total ticket price income of both normal and on-demand service (RMB).

.

It is evident that the number of buses on Route a and Route b increases by one during the rush hour of 7:00–8:00 in the case of the on-demand bus mode compared to the case of the without on-demand service transit mode. This increase is due to the consideration of travelers’ time cost and the provision of on-demand services, resulting in a denser departure frequency than the case of the without on-demand service transit mode. Consequently, the number of vehicles has increased to minimize the travel time of all travelers.

5.3. The Total Generalized Cost

Figure 9 presents the total generalized costs incurred by both bus companies and travelers in two scenarios: without on-demand service transit mode and with the on-demand bus mode. During the rush hour period of 7:00–8:00, the total generalized cost is 2672.6 RMB without on-demand service transit mode, while with on-demand bus mode, it is 1861.3 RMB. This indicates a reduction of 30.36% (811.3 RMB) in the total generalized cost as compared to without on-demand service transit mode. Additionally, during the off-rush hour period of 11:00–12:00, the total generalized cost is 1275.1 RMB with on-demand bus mode, which is 15.35% (231.1 RMB) lower than the cost without on-demand service transit mode. Hence, it can be observed that the proposed on-demand bus mode has the potential to reduce the total generalized cost during both rush and off-rush hours. Furthermore, the decrease in total generalized cost validates the effectiveness of the proposed bi-level model.

5.4. Travelers’ Time Cost

Figure 10 depicts the time costs incurred by all travelers in the scenarios of without on-demand service transit mode and with the on-demand bus mode. During the rush hour from 7:00 to 8:00, the time cost of all travelers in the on-demand bus mode is 1566.3 RMB, which is 37.08% lower than that in the case without on-demand service transit mode. Furthermore, during the off-rush hour from 11:00 to 12:00, despite the decrease in the percentage of travelers selecting the on-demand stop, the time cost of all travelers still reduces to 1054 RMB, representing a 20.33% decrease compared to the case without on-demand service transit mode. The analysis indicates that the on-demand bus mode, which considers the walking and waiting time of travelers, can effectively reduce the travel time of all travelers and enhance the bus service level in remote areas. Additionally, the reduction in the time cost confirms the effectiveness of the on-demand bus mode in remote areas.

Table 6. Waiting times at different stops in different time windows.

Stop	Without on-Demand Service Transit Mode		On-Demand Bus Mode
	7:00–8:00	11:00–12:00	7:00–8:00	11:00–12:00
Normal stops of route a	15	15	7.5	12.5
Normal stops of route b	20	20	10	15
On-demand stop	n.a.	n.a.	4.3	6.8

displays the average waiting times of travelers in the cases of the without on-demand service transit mode and on-demand bus mode. During 7:00–8:00 (rush hour), the average waiting times of travelers at normal stops of routes a and b are 7.5 and 10 min, respectively, in the case of the on-demand bus mode. This represents a 50% reduction in average waiting time compared to without on-demand service transit mode in both routes. Even during 11:00–12:00 (off-rush hour), the average waiting time of travelers at normal stops in the case of the on-demand bus mode decreases by 16.67% on route a and 25% on route b. This is because the added on-demand departures result in a denser optimal departure frequency, which reduces the average waiting time of travelers at normal stops.

In addition, during 7:00–8:00 (rush hour) and 11:00–12:00 (off-rush hour), the average waiting times of travelers who choose to board at on-demand stops are only 4.3 and 6.8 min, respectively. This is because when the destination of a traveler is the stop on overlapped routes, the traveler can be served by both bus routes at an on-demand stop, which effectively reduces the waiting time at on-demand stops.

5.5. Revenue

The revenue earned by bus companies during the rush hour (7:00–8:00) and off-rush hour (11:00–12:00) in the cases of without on-demand service transit mode and on-demand bus mode are presented in Figure 11. The ticket price for travelers boarding at an on-demand stop is 3.6 RMB. During rush hour, the revenue in the case of on-demand bus mode is 379.9 RMB, which is 15.89% (52.1 RMB) more than that in the without case. In the off-rush hour, the revenue is 140.0 RMB, which is 61.24% (53.2 RMB) more than that in the without case. Therefore, it can be observed that the proposed on-demand bus mode not only reduces the total generalized cost but also increases the revenue of bus companies.

5.6. Discussion

This paper optimizes bus services in remote areas, proposes on-demand services to meet the diverse travel needs of residents in remote areas. This research has found that on the one hand, travelers in remote areas are more inclined to choose an on-demand service, which is consistent with previous research results. Travelers are more willing to pay higher ticket price than normal bus services to obtain more flexible travel services to meet travel demands [49]. This article aims to optimize bus services with partially overlapping routes in remote areas with the goal of minimizing the generalized cost, so that the optimized ticket prices of on-demand service are not significantly different from ticket prices of normal service. However, the waiting time at on-demand stop is significantly reduced, which leads to more travelers choosing the on-demand service. On the other hand, the on-demand bus mode can reduce the travel time of residents in remote areas, improve the public transportation service level, and improve the travel conditions of residents. This further verifies the applicability of the on-demand bus mode in low population density areas [50].

6. Conclusions

This paper proposes an on-demand bus mode to improve bus service quality in remote areas, taking into consideration the attributes of dispersed travel demand, time-sensitive travelers, and those with urgent travel needs. A bi-level model is constructed to optimize bus departure frequency and ticket price of on-demand service, with the objective of minimizing the total generalized cost, which includes the bus operating cost and the value of travelers’ travel time. A case study is conducted using actual data from Meishan Island (Ningbo), and the results show that:

(1)

Supplying on-demand service can reduce total generalized costs by 30.36% and 15.35% in rush and off-rush hours, respectively, with the value of travelers’ time decreasing by 37.08% and 20.33%. This indicates that on-demand bus modes can effectively re-duce the travel time and improve bus service quality in remote areas when considering the value of travelers’ time cost.

(2)

Travelers are inclined to choose on-demand stops in both rush and off-rush hours. The on-demand bus mode not only reduces the total generalized cost of bus operation and traveler time but also satisfies a diversified travel demand and enriches the bus service.

(3)

Bus companies can improve service quality in remote areas and generate additional revenue by taking travelers’ time cost into account when providing bus services.

Based on the above findings, it can be concluded that implementing on-demand bus service in remote areas not only reduces the travel time of local residents, improving their living standards, but also meets the travel needs of travelers seeking low-cost and efficient travel options, providing high-quality public transport services to residents in remote areas. Therefore, the government and bus companies could consider residents’ travel time and bus operating costs holistically, and optimize bus services in remote areas based on the unique needs of local residents across different regions and different time windows. This will effectively meet the diverse needs of travelers, enhance residents’ happiness, and increase social welfare.

When optimizing bus services in remote areas, this paper assumes that the total number of bus trips taken by residents before and after optimization remains constant, without considering the potential impact of changes in bus service modes on residents’ choice of transportation mode (i.e., bus or car), which may lead to changes in the total number of bus trips taken by residents. Therefore, analyzing the impact of on-demand bus services on residents’ transportation mode choices and developing bus operation modes that better meet the actual needs of residents are important future research directions.